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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>According to the theorem,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_5.html">
\begin{equation*}
y_2(x)=\sum_{n=0}^{\infty} a_n (x-2)^n,
\end{equation*}
</div>
<p class="continuation">with <span class="process-math">\(a_0=0,a_1=1.\)</span> In (<a href="" class="xref" data-knowl="./knowl/eq5_5.html" title="Equation 5.3.2">(5.3.2)</a>), let <span class="process-math">\(a_0=0, a_1=1\text{.}\)</span> Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_5.html">
\begin{equation*}
\begin{aligned}
&amp;n=0: 2 \cdot 2 \cdot 1 \cdot a_2+3 \cdot 1 \cdot a_1=0 \to a_2=-\frac{3}{4},\\
&amp;n=1: 2 \cdot 3 \cdot 2 \cdot a_3+ 3 \cdot 2 \cdot \left(-\frac{3}{4}\right)+4 \cdot 1=0 \to a_3=\frac{1}{24},\\
&amp;n=2: 2 \cdot 4 \cdot 3 \cdot a_4+3 \cdot 3 \cdot \frac{1}{24}+5 \cdot \left(-\frac{3}{4}\right)=0 \to a_4=\frac{9}{64}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_5.html">
\begin{equation*}
y_2(x)=(x-2)-\frac{3}{4}(x-2)^2+\frac{1}{24} (x-2)^3+
\frac{9}{64}(x-4)^4+\cdots.
\end{equation*}
</div>
<span class="incontext"><a href="sec5_3.html#p-219" class="internal">in-context</a></span>
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